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Mar
15
comment Expectation of the difference of sums
@Michael: You average over all permutations. That erases all distinctions between the entries. The only difference that still matters is whether a term is the product of two identical entries or two different entries. If it's the product of two different entries, the permutations will permute it into all possible pairs of different entries, and the number of permutations permuting it into a given pair will be the same for all pairs of different entries. If this isn't obvious, I suggest to try it out with a small number like $m=2$. And you're welcome :-).
Mar
15
comment What equity is necessary to offer the doubling cube in Backgammon (dice game)
Yes, you can gain an advantage against a suboptimal strategy that way, but the equilibrium strategy must have both thresholds the same, so to find the equilibrium it should suffice to test all strategies with equal thresholds against each other, no?
Mar
15
answered Eigenvector corresponding to zero eigenvalue / identical eigenvalues, not-identical eigenvectors
Mar
15
comment Sum of three consecutive cubes
About that last solution: That's nice; it contains the cubes for the Hardy–Ramanujan number :-)
Mar
15
comment Sum of three consecutive cubes
Shouldn't it be $6a$ instead of $6a^2$?
Mar
15
comment What equity is necessary to offer the doubling cube in Backgammon (dice game)
This is a very nice answer; I enjoyed reading it :-). I was surprised by the $80\%$ result; I didn't expect it to be that high; so I wrote a simulation to test it. The result was that indeed the $80\%$ strategy wins against other strategies. Against the strategy of doubling whenever one is in the lead, it wins more than $1.5$ as many points as the opponent. Here's the code.
Mar
15
comment How do you pronounce (partial) derivatives?
See math.stackexchange.com/questions/110565/…, though I don't think it's an exact duplicate.
Mar
15
comment Characterization of Regulated Functions
@ItsNotObvious: You're welcome.
Mar
15
comment How many sequence of integers ($j_1 , j_2 , . . . , j_k$) are there such that $0 ≤ j_1 ≤ j_2 ≤ . . . ≤ j_k ≤ n$?
See also math.stackexchange.com/questions/119144/….
Mar
15
comment calculating lottery odds for non-descending order
See also math.stackexchange.com/questions/117835/….
Mar
15
revised Residue of $z^2 e^{1/\sin z}$ at $z=\pi$
added countour integration link
Mar
15
comment simple graph theory cycle problem
@Marc: I thought your point about "at least three vertices" was based on the idea that there's a cycle between two adjacent vertices that uses each of the vertices only once but uses the edge between them twice. So I was thinking that this problem is excluded if a cycle is required to be edge-disjoint. Perhaps I misunderstood you?
Mar
15
comment Designing an efficient sampling strategy
I think you'll have to take care not to introduce bias by having the sampling strategy depend on the initial samples. There might be ways to avoid that, but to be on the safe side you could first do some sampling to decide the strategy and then not use those initial samples for the estimate.
Mar
15
comment simple graph theory cycle problem
@Marc: I understand your first point now. On your second point, though, it seems that "simple cycle" is usually taken to mean not only vertex-disjoint but also edge-disjoint (for instance in Wikipedia); if I understand correctly, your second point was based on a definition of a simple cycle as vertex-disjoint but not necessarily edge-disjoint?
Mar
15
answered Residue of $z^2 e^{1/\sin z}$ at $z=\pi$
Mar
15
comment simple graph theory cycle problem
@Marc: I don't understand. Aren't a connected undirected graph without simple cycles and a connected undirected graph without general cycles the same thing? Why does that quote tell us anything about how "cycle" is used?
Mar
15
answered Coefficient of $z^{n-1}$ term in a generating function
Mar
15
answered Density of truncated normal distribution?
Mar
15
comment Distribute $N$ objects to $K$ boxes such that no box has more than $c$ objects in it
@user9915: Have you followed the link to the Wikipedia article on inclusion-exclusion?
Mar
15
comment what are the applications of the isomorphic graphs?
The term is "NP-complete", not "NP complex".